You would need a car -- although getting a taxi to take you on a tour up the mountain is a possibility,driving around the North side and Skyline drive stopping here and there for pictures, there is a lot of beauty on STT that a trip between the airport and ferry dock,, even the Red Hook trip doesn't do justice to the sights from the heights.
The scenic overlooks at Louisehoj, Drakes Seat even Mountaintop or Paradise Point are worthy, driving to the West end, visiting the Bordeaux farmers market, and coming down Blackpoint Hill also has a few overlooks on the way to the airport.
Kevin Campbell at the Tortola ferry dock is a great taxi resource . Rachel Brown is another.
without a car or taxi:
You could also contact the St Thomas Historical Society and see if you can get a walking tour of town, the unique architecture of town is interesting to many people and the guides/docents are excellent.